WEBVTT 00:00:08.280 --> 00:00:09.650 Okay. 00:00:09.650 --> 00:00:15.450 This lecture is mostly about the idea of similar matrixes. 00:00:15.450 --> 00:00:18.520 I'm going to tell you what that word similar means 00:00:18.520 --> 00:00:22.990 and in what way two matrixes are called similar. 00:00:22.990 --> 00:00:25.870 But before I do that, I have a little more 00:00:25.870 --> 00:00:28.310 to say about positive definite matrixes. 00:00:28.310 --> 00:00:34.880 You can tell this is a subject I think is really important and I 00:00:34.880 --> 00:00:39.170 told you what positive definite meant -- 00:00:39.170 --> 00:00:41.430 it means that this -- 00:00:41.430 --> 00:00:45.490 this expression, this quadratic form, x transpose I 00:00:45.490 --> 00:00:48.010 x is always positive. 00:00:48.010 --> 00:00:51.710 But the direct way to test it was with eigenvalues 00:00:51.710 --> 00:00:52.940 or pivots or determinants. 00:00:55.850 --> 00:00:58.730 So I -- we know what it means, we know how to test it, 00:00:58.730 --> 00:01:03.820 but I didn't really say where positive definite matrixes come 00:01:03.820 --> 00:01:05.140 from. 00:01:05.140 --> 00:01:10.470 And so one thing I want to say is that they come from least 00:01:10.470 --> 00:01:15.780 squares in -- and all sorts of physical problems start with 00:01:15.780 --> 00:01:19.520 a rectangular matrix -- well, you remember in least squares 00:01:19.520 --> 00:01:23.810 the crucial combination was A transpose A. 00:01:23.810 --> 00:01:27.970 So I want to show that that's a positive definite matrix. 00:01:27.970 --> 00:01:28.727 Can -- so I -- 00:01:28.727 --> 00:01:31.060 I'm going to speak a little more about positive definite 00:01:31.060 --> 00:01:34.730 matrixes, just recapping -- 00:01:34.730 --> 00:01:38.400 so let me ask a question. 00:01:38.400 --> 00:01:40.910 It may be on the homework. 00:01:40.910 --> 00:01:45.330 Suppose a matrix A is positive definite. 00:01:45.330 --> 00:01:48.360 I mean by that it's all -- 00:01:48.360 --> 00:01:50.100 I'm assuming it's symmetric. 00:01:50.100 --> 00:01:52.760 That's always built into the definition. 00:01:52.760 --> 00:01:56.440 So we have a symmetric positive definite matrix. 00:01:56.440 --> 00:01:58.610 What about its inverse? 00:01:58.610 --> 00:02:03.180 Is the inverse of a symmetric positive definite matrix also 00:02:03.180 --> 00:02:06.310 symmetric positive definite? 00:02:06.310 --> 00:02:09.789 So you quickly think, okay, what do I 00:02:09.789 --> 00:02:15.240 know about the pivots of the inverse matrix? 00:02:15.240 --> 00:02:17.390 Not much. 00:02:17.390 --> 00:02:19.620 What do I know about the eigenvalues 00:02:19.620 --> 00:02:22.540 of the inverse matrix? 00:02:22.540 --> 00:02:24.480 Everything, right? 00:02:24.480 --> 00:02:27.310 The eigenvalues of the inverse are 00:02:27.310 --> 00:02:31.290 one over the eigenvalues of the matrix. 00:02:31.290 --> 00:02:34.680 So if my matrix starts out positive definite, 00:02:34.680 --> 00:02:39.490 then right away I know that its inverse is positive definite, 00:02:39.490 --> 00:02:42.440 because those positive eigenvalues -- 00:02:42.440 --> 00:02:47.560 then one over the eigenvalue is also positive. 00:02:47.560 --> 00:02:51.820 What if I know that A -- a matrix A and a matrix B are 00:02:51.820 --> 00:02:53.280 both positive definite? 00:02:53.280 --> 00:02:54.780 But let me ask you this. 00:02:54.780 --> 00:03:04.140 Suppose if A and B are positive definite, what about -- 00:03:04.140 --> 00:03:05.940 what about A plus B? 00:03:09.020 --> 00:03:11.600 In some way, you hope that that would be true. 00:03:11.600 --> 00:03:15.730 It's -- positive definite for a matrix is kind of like positive 00:03:15.730 --> 00:03:18.540 for a real number. 00:03:18.540 --> 00:03:22.300 But we don't know the eigenvalues of A plus B. 00:03:22.300 --> 00:03:24.860 We don't know the pivots of A plus B. 00:03:24.860 --> 00:03:27.870 So we just, like, have to go down this list of, all right, 00:03:27.870 --> 00:03:31.130 which approach to positive definite 00:03:31.130 --> 00:03:33.050 can we get a handle on? 00:03:33.050 --> 00:03:35.570 And this is a good one. 00:03:35.570 --> 00:03:36.670 This is a good one. 00:03:36.670 --> 00:03:39.190 Can we -- how would we decide that -- 00:03:39.190 --> 00:03:43.610 if A was like this and if B was like this, 00:03:43.610 --> 00:03:48.410 then we would look at x transpose A plus B x. 00:03:48.410 --> 00:03:51.280 I'm sure this is in the homework. 00:03:51.280 --> 00:03:55.320 Now -- so we have x transpose A x bigger than zero, 00:03:55.320 --> 00:03:59.940 x transpose B x positive for all -- for all x, 00:03:59.940 --> 00:04:02.180 so now I ask you about this 00:04:02.180 --> 00:04:03.090 guy. 00:04:03.090 --> 00:04:06.800 And of course, you just add that and that 00:04:06.800 --> 00:04:09.940 and we get what we want. 00:04:09.940 --> 00:04:14.320 If A and B are positive definites, so is A plus B. 00:04:14.320 --> 00:04:15.810 So that's what I've shown. 00:04:15.810 --> 00:04:20.910 So is A plus B. 00:04:20.910 --> 00:04:27.310 Just -- be sort of ready for all the approaches through 00:04:27.310 --> 00:04:31.260 eigenvalues and through this expression. 00:04:31.260 --> 00:04:35.500 And now, finally, one more thought about positive definite 00:04:35.500 --> 00:04:39.020 is this combination that came up in least squares. 00:04:39.020 --> 00:04:41.260 Can I do that? 00:04:41.260 --> 00:04:48.290 So now -- now suppose A is rectangular, m by n. 00:04:52.890 --> 00:04:58.380 I -- so I'm sorry that I've used the same letter A 00:04:58.380 --> 00:05:02.730 for the positive definite matrixes in the eigenvalue 00:05:02.730 --> 00:05:07.660 chapter that I used way back in earlier chapters when 00:05:07.660 --> 00:05:09.620 the matrix was rectangular. 00:05:09.620 --> 00:05:12.750 Now, that matrix -- a rectangular matrix, 00:05:12.750 --> 00:05:14.569 no way its positive definite. 00:05:14.569 --> 00:05:15.360 It's not symmetric. 00:05:18.240 --> 00:05:20.450 It's not even square in general. 00:05:20.450 --> 00:05:25.540 But you remember that the key for these rectangular ones 00:05:25.540 --> 00:05:28.147 was A transpose A. 00:05:28.147 --> 00:05:28.730 That's square. 00:05:34.922 --> 00:05:35.630 That's symmetric. 00:05:38.690 --> 00:05:41.310 Those are things we knew -- 00:05:41.310 --> 00:05:44.890 we knew back when we met this thing 00:05:44.890 --> 00:05:49.020 in the least square stuff, in the projection stuff. 00:05:49.020 --> 00:05:52.340 But now we know something more -- 00:05:52.340 --> 00:05:56.450 we can ask a more important question, a deeper question -- 00:05:56.450 --> 00:05:57.920 is it positive definite? 00:06:01.110 --> 00:06:02.770 And we sort of hope so. 00:06:02.770 --> 00:06:04.730 Like, we -- we might -- 00:06:04.730 --> 00:06:09.200 in analogy with numbers, this is like -- 00:06:09.200 --> 00:06:15.360 sort of like the square of a number, and that's positive. 00:06:15.360 --> 00:06:19.930 So now I want to ask the matrix question. 00:06:19.930 --> 00:06:24.350 Is A transpose A positive definite? 00:06:24.350 --> 00:06:27.740 Okay, now it's -- so again, it's a rectangular A that 00:06:27.740 --> 00:06:31.790 I'm starting with, but it's the combination A transpose A 00:06:31.790 --> 00:06:36.150 that's the square, symmetric and hopefully positive definite 00:06:36.150 --> 00:06:36.780 matrix. 00:06:36.780 --> 00:06:41.700 So how -- how do I see that it is positive definite, 00:06:41.700 --> 00:06:43.750 or at least positive semi-definite? 00:06:43.750 --> 00:06:44.440 You'll see that. 00:06:47.180 --> 00:06:52.280 Well, I don't know the eigenvalues of this product. 00:06:52.280 --> 00:06:54.250 I don't want to work with the pivots. 00:06:54.250 --> 00:07:00.060 The right thing -- the right quantity to look at is this, 00:07:00.060 --> 00:07:01.800 x transpose Ax -- 00:07:01.800 --> 00:07:07.450 A -- x transpose times my matrix times x. 00:07:07.450 --> 00:07:09.280 I'd like to see that this thing -- 00:07:09.280 --> 00:07:13.450 that that expression is always positive. 00:07:13.450 --> 00:07:16.310 I'm not doing it with numbers, I'm doing it with symbols. 00:07:16.310 --> 00:07:21.890 Do you see -- how do I see that that expression comes out 00:07:21.890 --> 00:07:23.620 positive? 00:07:23.620 --> 00:07:29.260 I'm taking a rectangular matrix A and an A transpose -- 00:07:29.260 --> 00:07:31.270 that gives me something square symmetric, 00:07:31.270 --> 00:07:34.030 but now I want to see that if I multiply -- 00:07:34.030 --> 00:07:34.970 that if I do this -- 00:07:34.970 --> 00:07:38.140 I form this quadratic expression that I 00:07:38.140 --> 00:07:42.160 get this positive thing that goes upwards when I graph it. 00:07:42.160 --> 00:07:44.210 How do I see that that's positive, 00:07:44.210 --> 00:07:47.860 or absolutely it isn't negative anyway? 00:07:47.860 --> 00:07:51.990 We'll have to, like, spend a minute on the question 00:07:51.990 --> 00:07:54.620 could it be zero, but it can't be negative. 00:07:54.620 --> 00:07:56.310 Why can this never be negative? 00:07:59.060 --> 00:08:02.590 The argument is -- 00:08:02.590 --> 00:08:08.930 like the one key idea in so many steps in linear algebra -- 00:08:08.930 --> 00:08:12.960 put those parentheses in a good way. 00:08:12.960 --> 00:08:19.560 Put the parentheses around Ax and what's the first part? 00:08:19.560 --> 00:08:22.280 What's this x transpose A transpose? 00:08:22.280 --> 00:08:25.540 That is Ax transpose. 00:08:29.000 --> 00:08:30.810 So what do we have? 00:08:30.810 --> 00:08:35.650 We have the length squared of Ax. 00:08:35.650 --> 00:08:39.909 We have -- that's the column vector Ax that's the row vector 00:08:39.909 --> 00:08:48.380 Ax, its length squared, certainly greater than 00:08:48.380 --> 00:08:51.310 or possibly equal to zero. 00:08:51.310 --> 00:08:53.860 So we have to deal with this little possibility. 00:08:53.860 --> 00:08:55.070 Could it be equal? 00:08:55.070 --> 00:08:58.970 Well, when could the length squared be zero? 00:08:58.970 --> 00:09:02.320 Only if the vector is zero, right? 00:09:02.320 --> 00:09:08.340 That's the only vector that has length squared zero. 00:09:08.340 --> 00:09:11.080 So we have -- we would like to -- 00:09:11.080 --> 00:09:15.230 I would like to get that possibility out of there. 00:09:15.230 --> 00:09:18.920 So I want to have Ax never -- never be zero, 00:09:18.920 --> 00:09:21.340 except of course for the zero vector. 00:09:21.340 --> 00:09:24.280 How do I assure that Ax is never zero? 00:09:24.280 --> 00:09:27.650 The -- in other words, how do I show that there's no null space 00:09:27.650 --> 00:09:28.370 of A? 00:09:28.370 --> 00:09:33.420 The rank should be -- 00:09:33.420 --> 00:09:39.830 so now remember -- what's the rank when there's no null 00:09:39.830 --> 00:09:41.600 space? 00:09:41.600 --> 00:09:43.680 By no null space, you know what I mean. 00:09:43.680 --> 00:09:46.370 Only the zero vector in the null space. 00:09:46.370 --> 00:09:52.850 So if I have a -- if I have an 11 by 5 matrix -- 00:09:52.850 --> 00:09:59.230 so it's got 11 rows, 5 columns, when is there no null space? 00:09:59.230 --> 00:10:04.380 So the columns should be independent -- what's the rank? 00:10:04.380 --> 00:10:06.810 n 5 -- rank n. 00:10:06.810 --> 00:10:12.170 Independent columns, when -- so if I -- 00:10:12.170 --> 00:10:15.830 then I conclude yes, positive definite. 00:10:15.830 --> 00:10:19.020 And this was the assumption -- then A transpose A is 00:10:19.020 --> 00:10:23.210 invertible -- 00:10:23.210 --> 00:10:27.330 the least squares equations all work fine. 00:10:27.330 --> 00:10:33.870 And more than that -- the matrix is even positive definite. 00:10:33.870 --> 00:10:38.150 And I just to say one comment about numerical things, 00:10:38.150 --> 00:10:41.770 with a positive definite matrix, you never 00:10:41.770 --> 00:10:45.010 have to do row exchanges. 00:10:45.010 --> 00:10:50.230 You never run into unsuitably small numbers or zeroes 00:10:50.230 --> 00:10:51.560 in the pivot position. 00:10:51.560 --> 00:10:54.960 They're the right -- they're the great matrixes to compute with, 00:10:54.960 --> 00:10:57.250 and they're the great matrixes to study. 00:10:57.250 --> 00:11:05.900 So that's -- I wanted to take this first ten minutes of grab 00:11:05.900 --> 00:11:12.150 the first ten minutes away from similar matrixes and continue 00:11:12.150 --> 00:11:14.710 a -- this much more with positive definite. 00:11:14.710 --> 00:11:19.270 I'm really at this point, now, coming close 00:11:19.270 --> 00:11:24.300 to the end of the heart of linear algebra. 00:11:24.300 --> 00:11:28.160 The positive definiteness brought everything together. 00:11:28.160 --> 00:11:32.920 Similar matrixes, which is coming the rest of this hour 00:11:32.920 --> 00:11:38.400 is a key topic, and please come on Monday. 00:11:38.400 --> 00:11:42.690 Monday is about what's called the SVD, singular values. 00:11:42.690 --> 00:11:49.340 It's the -- has become a central fact in -- 00:11:49.340 --> 00:11:53.530 a central part of linear algebra. 00:11:53.530 --> 00:11:57.940 I mean, you can come after Monday also, but -- 00:11:57.940 --> 00:12:04.800 Monday is, -- that singular value thing has made it 00:12:04.800 --> 00:12:05.990 into this course. 00:12:05.990 --> 00:12:08.870 Ten years ago, five years ago it wasn't in the course, 00:12:08.870 --> 00:12:10.640 now it has to be. 00:12:10.640 --> 00:12:11.490 Okay. 00:12:11.490 --> 00:12:16.560 So can I begin today's lecture proper with this idea 00:12:16.560 --> 00:12:18.610 of similar matrixes. 00:12:18.610 --> 00:12:22.220 This is what similar matrixes mean. 00:12:22.220 --> 00:12:23.680 So here -- let's start again. 00:12:23.680 --> 00:12:24.720 I'll write it again. 00:12:24.720 --> 00:12:27.690 So A and B are similar. 00:12:27.690 --> 00:12:33.550 A and B are -- now I'm -- these matrixes -- 00:12:33.550 --> 00:12:38.720 I'm no longer talking about symmetric matrixes, in -- 00:12:38.720 --> 00:12:42.750 at least no longer expecting symmetric matrixes. 00:12:42.750 --> 00:12:46.960 I'm talking about two square matrixes n by n. 00:12:46.960 --> 00:12:51.790 A and B, they're n by n matrixes. 00:12:54.760 --> 00:12:59.110 And I'm introducing this word similar. 00:12:59.110 --> 00:13:00.880 So I'm going to say what does it mean? 00:13:00.880 --> 00:13:08.520 It means that they're connected in the way -- 00:13:08.520 --> 00:13:12.920 well, in the way I've written here, so let me rewrite it. 00:13:12.920 --> 00:13:24.260 That means that for some matrix M, which has to be invertible, 00:13:24.260 --> 00:13:26.140 because you'll see that -- 00:13:26.140 --> 00:13:29.680 this one matrix is -- 00:13:29.680 --> 00:13:32.980 take the other matrix, multiply on the right 00:13:32.980 --> 00:13:39.700 by M and on the left by M inverse. 00:13:39.700 --> 00:13:43.810 So the question is, why that combination? 00:13:43.810 --> 00:13:46.890 But part of the answer you know already. 00:13:46.890 --> 00:13:52.070 You remember -- we've done this -- we've taken a matrix A -- 00:13:52.070 --> 00:13:58.620 so let's do an example of similar. 00:13:58.620 --> 00:14:06.340 Suppose A -- the matrix A -- suppose it has a full set 00:14:06.340 --> 00:14:09.090 of eigenvectors. 00:14:09.090 --> 00:14:12.910 They go in this eigenvector matrix S. 00:14:12.910 --> 00:14:15.220 Then what was the main point of the whole -- 00:14:15.220 --> 00:14:19.960 the main calculation of the whole chapter was -- is -- 00:14:19.960 --> 00:14:25.380 use that eigenvector matrix S and its inverse 00:14:25.380 --> 00:14:33.720 comes over there to produce the nicest possible matrix lambda. 00:14:33.720 --> 00:14:38.160 Nicest possible because it's diagonal. 00:14:38.160 --> 00:14:47.615 So in our new language, this is saying A is similar to lambda. 00:14:51.440 --> 00:14:56.440 A is similar to lambda, because there is a matrix, 00:14:56.440 --> 00:14:58.970 and this particular -- 00:14:58.970 --> 00:15:01.140 there is an M and this particular M 00:15:01.140 --> 00:15:05.220 is this important guy, this eigenvector matrix. 00:15:05.220 --> 00:15:13.040 But if I take a different matrix M and I look at M inverse A M, 00:15:13.040 --> 00:15:15.350 the result won't come out diagonal, 00:15:15.350 --> 00:15:21.140 but it will come out a matrix B that's similar to A. 00:15:21.140 --> 00:15:25.870 Do you see that I'm -- what I'm doing is, like -- 00:15:25.870 --> 00:15:28.700 I'm putting these matrixes into families. 00:15:28.700 --> 00:15:34.060 All the matrixes in one -- in the family are similar to each 00:15:34.060 --> 00:15:35.070 other. 00:15:35.070 --> 00:15:39.080 They're all -- each one in this family is connected to each 00:15:39.080 --> 00:15:42.550 other one by some matrix M and the -- 00:15:42.550 --> 00:15:47.660 like the outstanding member of the family is the diagonal guy. 00:15:47.660 --> 00:15:50.700 I mean, that's the simplest, neatest matrix 00:15:50.700 --> 00:15:55.680 in this family of all the matrixes that are similar to A, 00:15:55.680 --> 00:15:58.530 the best one is lambda. 00:15:58.530 --> 00:16:01.720 But there are lots of others, because I can take different -- 00:16:01.720 --> 00:16:04.870 instead of S, I can take any old matrix M, 00:16:04.870 --> 00:16:08.440 any old invertible matrix and -- and do it. 00:16:08.440 --> 00:16:10.020 I'd better do an example. 00:16:10.020 --> 00:16:10.520 Okay. 00:16:10.520 --> 00:16:16.901 Suppose I take A as the matrix two one one two. 00:16:16.901 --> 00:16:17.400 Okay. 00:16:21.510 --> 00:16:25.470 Do you know the eigenvalue matrix for that? 00:16:25.470 --> 00:16:28.630 The eigenvalues of that matrix are -- 00:16:28.630 --> 00:16:33.040 well, three and one. 00:16:33.040 --> 00:16:37.430 So that -- and the eigenvectors would be easy to find. 00:16:37.430 --> 00:16:40.780 So this matrix is similar to this one. 00:16:40.780 --> 00:16:43.080 But my point is -- 00:16:43.080 --> 00:16:49.410 but also, I can also take my matrix, two one one two, 00:16:49.410 --> 00:16:52.090 I could multiply it by -- let's see, what -- 00:16:52.090 --> 00:16:55.580 I'm just going to cook up a matrix M here. 00:16:55.580 --> 00:17:00.350 I'm -- I'll -- let me just invent -- one four one zero. 00:17:00.350 --> 00:17:02.710 And over here I'll put M inverse, 00:17:02.710 --> 00:17:05.839 and because I happened to make that triangular, 00:17:05.839 --> 00:17:09.940 I know that its inverse is that, right? 00:17:09.940 --> 00:17:13.819 So there's M inverse A M, that's going to produce some matrix -- 00:17:13.819 --> 00:17:17.200 oh, well, I've got to do the multiplication, 00:17:17.200 --> 00:17:19.380 so hang on a second, let -- 00:17:19.380 --> 00:17:22.880 I'll just copy that one minus four zero one 00:17:22.880 --> 00:17:33.930 and multiply these guys so I'm getting two nine one and six, 00:17:33.930 --> 00:17:36.010 I think. 00:17:36.010 --> 00:17:39.680 Can you check it as I go, because you -- see I'm just -- 00:17:39.680 --> 00:17:43.850 so that's two minus four, I'm getting a minus two nine 00:17:43.850 --> 00:17:48.550 minus 24 is a minus 15, my God, how did I get this? 00:17:48.550 --> 00:17:50.935 And that's probably one and six. 00:17:54.560 --> 00:17:57.810 So there's my matrix B. 00:17:57.810 --> 00:18:02.180 And there's my matrix lambda, there's my matrix A 00:18:02.180 --> 00:18:04.715 and my point is these are all similar matrixes. 00:18:07.250 --> 00:18:09.880 They all have something in common, 00:18:09.880 --> 00:18:13.130 besides being just two by two. 00:18:13.130 --> 00:18:17.190 They have something in common. 00:18:17.190 --> 00:18:21.310 And that's -- and what is it? 00:18:21.310 --> 00:18:26.540 What's the point about two matrixes that are built out 00:18:26.540 --> 00:18:27.840 of -- 00:18:27.840 --> 00:18:32.440 the B is built out of M inverse A M. 00:18:32.440 --> 00:18:34.840 What is it that A and B have in common? 00:18:34.840 --> 00:18:38.127 That's the main -- now I'm telling you the main fact about 00:18:38.127 --> 00:18:38.835 similar matrixes. 00:18:41.590 --> 00:18:44.910 They have the same eigenvalues. 00:18:44.910 --> 00:18:47.550 This is -- this chapter is about eigenvalues, 00:18:47.550 --> 00:18:51.350 and that's why we're interested in this family of matrixes that 00:18:51.350 --> 00:18:53.140 have the same eigenvalues. 00:18:53.140 --> 00:18:56.640 What are the eigenvalues in this example? 00:18:56.640 --> 00:18:57.570 Lambda. 00:18:57.570 --> 00:19:01.070 The eigenvalues of that I could compute. 00:19:01.070 --> 00:19:06.370 The eigenvalues of that I can compute really fast. 00:19:06.370 --> 00:19:10.330 So the eigenvalues are three and one -- 00:19:10.330 --> 00:19:11.940 for this for sure. 00:19:11.940 --> 00:19:15.520 Now did we -- do you see why the eigenvalues are three and one 00:19:15.520 --> 00:19:17.930 for that one? 00:19:17.930 --> 00:19:21.870 If I tell you the eigenvalues are three and one, you prick -- 00:19:21.870 --> 00:19:26.190 quickly process the trace, which is -- and four -- 00:19:26.190 --> 00:19:29.510 agrees with four and you process the determinant, 00:19:29.510 --> 00:19:31.990 three times one -- 00:19:31.990 --> 00:19:35.930 the determinant is three and you say yes, it's right. 00:19:35.930 --> 00:19:39.540 Now I'm hoping that the eigenvalues of this thing 00:19:39.540 --> 00:19:41.770 are three and one. 00:19:41.770 --> 00:19:45.450 May I process the trace and the determinant for that one? 00:19:45.450 --> 00:19:47.830 What's the trace here? 00:19:47.830 --> 00:19:52.510 The trace of this matrix is four minus two and six, 00:19:52.510 --> 00:19:54.320 and that's what it should be. 00:19:54.320 --> 00:19:59.210 What's the determinant minus twelve plus fifteen is three. 00:19:59.210 --> 00:20:00.390 The determinant is three. 00:20:00.390 --> 00:20:04.190 The eigenvalues of that matrix are also three and one. 00:20:04.190 --> 00:20:07.460 And you see I created this matrix just like -- 00:20:07.460 --> 00:20:11.600 I just took any M, like, one that popped into my head 00:20:11.600 --> 00:20:16.030 and computed M inverse A M, got that matrix, 00:20:16.030 --> 00:20:22.980 it didn't look anything special but it's -- 00:20:22.980 --> 00:20:26.290 like A itself, it has those eigenvalues three and one. 00:20:26.290 --> 00:20:31.370 So that's the main fact and let me write it down. 00:20:31.370 --> 00:20:45.290 Similar matrixes have the same eigenvalues. 00:20:45.290 --> 00:20:51.590 So I'll just put that as an important point. 00:20:51.590 --> 00:20:53.890 And think why. 00:20:56.690 --> 00:20:57.430 Why is that? 00:20:57.430 --> 00:21:01.120 So that's what that family of matrixes is. 00:21:01.120 --> 00:21:03.940 The matrixes that are similar to this A 00:21:03.940 --> 00:21:09.610 here are all the matrixes with eigenvalues three and one. 00:21:09.610 --> 00:21:12.450 Every matrix with eigenvalues three and one, 00:21:12.450 --> 00:21:16.870 there's some M that connects this guy 00:21:16.870 --> 00:21:19.810 to the one you think of. 00:21:19.810 --> 00:21:22.840 And then of course, the most special guy in the whole family 00:21:22.840 --> 00:21:26.690 is the diagonal one with eigenvalues three and one 00:21:26.690 --> 00:21:28.630 sitting there on the diagonal. 00:21:28.630 --> 00:21:30.590 But also, I could find -- 00:21:30.590 --> 00:21:34.130 I mean, tell me just a couple more members of the family. 00:21:34.130 --> 00:21:38.070 Another -- tell me another matrix that has eigenvalues 00:21:38.070 --> 00:21:38.680 three and one. 00:21:41.450 --> 00:21:44.740 Well, let's see, I -- oh, I'll just make it triangular. 00:21:47.610 --> 00:21:49.180 That's in the family. 00:21:49.180 --> 00:21:53.840 There is some M that -- that connects to this one. 00:21:53.840 --> 00:21:58.310 And -- and also this. 00:21:58.310 --> 00:22:02.630 There's some matrix M -- so that M inverse A M comes out to be 00:22:02.630 --> 00:22:03.180 that. 00:22:03.180 --> 00:22:06.290 There's a whole family here. 00:22:06.290 --> 00:22:11.710 And they all share the same eigenvalues. 00:22:11.710 --> 00:22:14.320 So why is that? 00:22:14.320 --> 00:22:14.930 Okay. 00:22:14.930 --> 00:22:21.880 I'm going to start -- the only possibility is to start with Ax 00:22:21.880 --> 00:22:24.180 equal lambda x. 00:22:24.180 --> 00:22:27.790 Okay, so suppose A has the eigenvalue lambda. 00:22:30.970 --> 00:22:33.880 Now I want to get B into the picture here somehow. 00:22:33.880 --> 00:22:37.930 You remember B is M inverse A M. 00:22:37.930 --> 00:22:39.910 Let's just remember that over here. 00:22:39.910 --> 00:22:44.390 B is M inverse A M. 00:22:44.390 --> 00:22:48.030 And I want to see its eigenvalues. 00:22:48.030 --> 00:22:51.520 How I going to get M inverse A M into this equation? 00:22:51.520 --> 00:22:54.830 Let me just sort of do it. 00:22:54.830 --> 00:22:59.970 I'll put an M times an M inverse in there, right? 00:22:59.970 --> 00:23:01.870 That was -- 00:23:01.870 --> 00:23:04.100 I haven't changed the left-hand side, 00:23:04.100 --> 00:23:06.420 so I better not change the right-hand side. 00:23:09.570 --> 00:23:12.850 So everybody's okay so far, I just put in there -- see, 00:23:12.850 --> 00:23:16.240 I want to get a -- so now I'll multiply on the left by M 00:23:16.240 --> 00:23:17.870 inverse -- 00:23:17.870 --> 00:23:20.200 I have to do the same to this side 00:23:20.200 --> 00:23:22.880 and that number lambda's just a number, 00:23:22.880 --> 00:23:25.380 so it factors out in the front. 00:23:25.380 --> 00:23:32.370 So what I have here is this was safe. 00:23:32.370 --> 00:23:34.760 I did the same thing to both sides. 00:23:34.760 --> 00:23:37.090 And now I've got B. 00:23:37.090 --> 00:23:38.270 There's B. 00:23:38.270 --> 00:23:42.210 That's B times this vector M inverse 00:23:42.210 --> 00:23:47.670 x is equal to lambda times this vector M inverse x. 00:23:47.670 --> 00:23:49.920 So what have I learned? 00:23:49.920 --> 00:23:53.980 I've learned that B times some vector 00:23:53.980 --> 00:23:55.600 is lambda times that vector. 00:23:55.600 --> 00:23:58.910 I've learned that lambda is an eigenvalue of B also. 00:23:58.910 --> 00:24:01.370 So this is -- if -- so this is -- 00:24:01.370 --> 00:24:05.600 if lambda's an eigenvalue of A, then I can write it this way 00:24:05.600 --> 00:24:10.110 and I discover that lambda's an eigenvalue of B. 00:24:10.110 --> 00:24:12.240 That's the end of the proof. 00:24:12.240 --> 00:24:16.670 The eigenvector didn't stay the same. 00:24:16.670 --> 00:24:19.410 Of course I don't expect the eigenvectors to stay the same. 00:24:19.410 --> 00:24:22.240 If all the eigenvalues are the same and all the eigenvectors 00:24:22.240 --> 00:24:25.120 are the same, then probably the matrix is the same. 00:24:27.960 --> 00:24:31.260 Here the eigenvector changes, so the eigenvector -- 00:24:31.260 --> 00:24:38.370 so the point is then the eigenvector of B -- 00:24:38.370 --> 00:24:44.880 of B is M inverse times the eigenvector of A. 00:24:44.880 --> 00:24:45.380 Okay. 00:24:51.989 --> 00:24:53.280 That's all that this says here. 00:24:53.280 --> 00:24:58.940 The eigenvector of A was X, and so the M inverse -- 00:24:58.940 --> 00:25:01.930 similar matrixes, then have the same eigenvalues 00:25:01.930 --> 00:25:05.070 and their eigenvectors are just moved around. 00:25:05.070 --> 00:25:08.790 Of course, that's what we -- that's what happened way back 00:25:08.790 --> 00:25:09.580 -- 00:25:09.580 --> 00:25:15.070 and the most important similar matrixes are to diagonalize. 00:25:15.070 --> 00:25:18.140 So what was the point when we diagonalized? 00:25:18.140 --> 00:25:20.830 The eigenvalues stayed the same, of course. 00:25:20.830 --> 00:25:22.380 Three and one. 00:25:22.380 --> 00:25:25.150 What about the eigenvectors? 00:25:25.150 --> 00:25:29.190 The eigenvectors were whatever they were for the matrix A, 00:25:29.190 --> 00:25:31.340 but then what were the eigenvectors 00:25:31.340 --> 00:25:34.630 for the diagonal matrix? 00:25:34.630 --> 00:25:37.370 They're just -- what are the eigenvectors of a diagonal 00:25:37.370 --> 00:25:37.900 matrix? 00:25:37.900 --> 00:25:41.130 They're just one zero and zero one. 00:25:41.130 --> 00:25:44.290 So this step made the eigenvectors nice, 00:25:44.290 --> 00:25:49.750 didn't change the eigenvalues, and every time we 00:25:49.750 --> 00:25:51.580 don't change the eigenvalues. 00:25:51.580 --> 00:25:53.021 Same eigenvalues. 00:25:53.021 --> 00:25:53.520 Okay. 00:25:56.150 --> 00:26:01.100 Now -- so I've got all these matrixes in -- 00:26:01.100 --> 00:26:07.380 I've got this family of matrixes with eigenvalues three and one. 00:26:07.380 --> 00:26:08.610 Fine. 00:26:08.610 --> 00:26:10.270 That's a nice family. 00:26:10.270 --> 00:26:15.520 It's nice because those two eigenvalues are different. 00:26:15.520 --> 00:26:17.280 I now have to -- 00:26:17.280 --> 00:26:19.460 to get into that -- 00:26:19.460 --> 00:26:25.320 the -- into the less happy possibility that the two 00:26:25.320 --> 00:26:26.960 eigenvalues could be the 00:26:26.960 --> 00:26:29.680 same. 00:26:29.680 --> 00:26:32.940 And then it's a little trickier, because you remember 00:26:32.940 --> 00:26:35.360 when two eigenvalues are the same, 00:26:35.360 --> 00:26:38.400 what's the bad possibility? 00:26:38.400 --> 00:26:42.030 That there might not be enough -- 00:26:42.030 --> 00:26:44.730 a full set of eigenvectors and we might not be able 00:26:44.730 --> 00:26:46.080 to diagonalize. 00:26:46.080 --> 00:26:50.550 So I need to discuss the bad case. 00:26:50.550 --> 00:26:53.070 So the bad -- can I just say bad? 00:26:57.500 --> 00:27:03.707 If lambda one equals lambda two, then the matrix 00:27:03.707 --> 00:27:04.873 might not be diagonalizable. 00:27:07.410 --> 00:27:11.010 Suppose lambda one equals lambda two equals four, 00:27:11.010 --> 00:27:11.510 say. 00:27:14.460 --> 00:27:20.990 Now if I look at the family of matrixes with eigenvalues four 00:27:20.990 --> 00:27:25.840 and four, well, one possibility occurs to me. 00:27:25.840 --> 00:27:38.120 One family with eigenvalues four and four has this matrix in it, 00:27:38.120 --> 00:27:41.420 four times the identity. 00:27:41.420 --> 00:27:46.400 Then another -- but now I want to ask also about the matrix 00:27:46.400 --> 00:27:48.930 four four one zero. 00:27:51.800 --> 00:27:55.290 And my point -- here's the whole point of this -- 00:27:55.290 --> 00:28:00.107 of this bad stuff, is that this guy is not in the same family 00:28:00.107 --> 00:28:00.690 with that one. 00:28:00.690 --> 00:28:05.310 The family of a -- of matrixes that have eigenvalues four 00:28:05.310 --> 00:28:08.760 and four is two families. 00:28:08.760 --> 00:28:16.081 There's this total loner here who's in a family off -- 00:28:16.081 --> 00:28:16.580 right? 00:28:16.580 --> 00:28:18.890 Just by himself. 00:28:18.890 --> 00:28:23.070 And all the others are in with this guy. 00:28:23.070 --> 00:28:35.090 So the big family includes this one. 00:28:35.090 --> 00:28:39.720 And it includes a whole lot of other matrixes, all -- 00:28:39.720 --> 00:28:44.310 in fact, in this two by two case, it -- you see where -- 00:28:44.310 --> 00:28:48.290 what do I mean -- so what I using, this word family -- 00:28:48.290 --> 00:28:51.980 in a family, I mean they're similar. 00:28:51.980 --> 00:28:56.570 So my point is that the only matrix that's similar to this 00:28:56.570 --> 00:28:58.840 is itself. 00:28:58.840 --> 00:29:02.240 The only matrix that's similar to four times the identity 00:29:02.240 --> 00:29:03.650 is four times the identity. 00:29:03.650 --> 00:29:05.320 It's off by itself. 00:29:05.320 --> 00:29:07.360 Why is that? 00:29:07.360 --> 00:29:11.750 The -- if this is my matrix, four times the identity, 00:29:11.750 --> 00:29:17.150 and I take it, I multiply on the right by any matrix M, 00:29:17.150 --> 00:29:22.430 I multiply on the left by M inverse, what do I get? 00:29:22.430 --> 00:29:28.720 This is any M, but what's the result? 00:29:28.720 --> 00:29:30.930 Well, factoring out a four, that's 00:29:30.930 --> 00:29:34.710 just the identity matrix in there. 00:29:34.710 --> 00:29:37.250 So then the M inverse cancels the M, 00:29:37.250 --> 00:29:39.420 so I've just got this matrix back again. 00:29:42.910 --> 00:29:45.310 So whatever the M is, I'm not getting 00:29:45.310 --> 00:29:47.610 any more members of the family. 00:29:47.610 --> 00:29:57.680 So this is one small family, because it only has one person. 00:29:57.680 --> 00:29:59.610 One matrix, excuse me. 00:29:59.610 --> 00:30:02.200 I think of these matrixes as people by this point, 00:30:02.200 --> 00:30:04.110 in eighteen oh six. 00:30:04.110 --> 00:30:08.990 Okay, the other family includes all the rest -- 00:30:08.990 --> 00:30:13.930 all other matrixes that have eigenvalues four and four. 00:30:13.930 --> 00:30:20.290 This is somehow the best one in that family. 00:30:20.290 --> 00:30:21.870 See, I can't make it diagonal. 00:30:21.870 --> 00:30:24.580 If I -- if it's diagonal, it's this one. 00:30:24.580 --> 00:30:27.020 It's in its own, by itself. 00:30:27.020 --> 00:30:29.250 So I have to think, okay, what's the nearest I 00:30:29.250 --> 00:30:32.480 can get to diagonal? 00:30:32.480 --> 00:30:36.380 But it will not be diagonalizable. 00:30:36.380 --> 00:30:39.670 That -- do you know that that matrix is not diagonalizable? 00:30:39.670 --> 00:30:42.050 Of course, because if it was diagonalizable, 00:30:42.050 --> 00:30:46.150 it would be similar to that, which it isn't. 00:30:46.150 --> 00:30:48.540 The eigenvalues of this are four and four, 00:30:48.540 --> 00:30:52.960 but what's the catch with that matrix? 00:30:52.960 --> 00:30:55.430 It's only got one eigenvector. 00:30:55.430 --> 00:30:57.560 That's a non-diagonalizable matrix. 00:30:57.560 --> 00:30:59.340 Only one eigenvector. 00:30:59.340 --> 00:31:06.700 And somehow, if I made that one into a ten or to a million, 00:31:06.700 --> 00:31:10.540 I could find an M, it's in the family, it's similar. 00:31:10.540 --> 00:31:14.570 But the best -- so the best guy in this family is this one. 00:31:14.570 --> 00:31:19.810 And this is called the Jordan -- 00:31:19.810 --> 00:31:24.800 so this guy Jordan picked out -- so he, like, studied, 00:31:24.800 --> 00:31:29.250 these families of matrixes, and each family, 00:31:29.250 --> 00:31:34.740 he picked out the nicest, most diagonal one. 00:31:34.740 --> 00:31:37.960 But not completely diagonal, because there's nobody -- 00:31:37.960 --> 00:31:40.430 there isn't a diagonal matrix in this family, 00:31:40.430 --> 00:31:44.780 so there's a one up there in the Jordan form. 00:31:44.780 --> 00:31:46.010 Okay. 00:31:46.010 --> 00:31:50.699 I think we've got to see some more matrixes in that family. 00:31:50.699 --> 00:31:52.990 So, all right, let me -- let's just think of some other 00:31:52.990 --> 00:31:58.570 matrixes whose eigenvalues are four and four but they're not 00:31:58.570 --> 00:32:01.220 four times the identity. 00:32:01.220 --> 00:32:03.130 So -- and I believe that -- 00:32:03.130 --> 00:32:06.660 that this -- that all the examples we pick up will be 00:32:06.660 --> 00:32:13.320 similar to each other and -- do you see why -- 00:32:13.320 --> 00:32:17.340 in this topic of similar matrixes, 00:32:17.340 --> 00:32:21.820 the climax is the Jordan form. 00:32:21.820 --> 00:32:23.910 So it says that every matrix -- 00:32:23.910 --> 00:32:29.310 I'll write down what the Jordan form -- what Jordan discovered. 00:32:29.310 --> 00:32:35.030 He found the best looking matrix in each family. 00:32:35.030 --> 00:32:41.080 And that's -- then we've got -- then we've covered all matrixes 00:32:41.080 --> 00:32:44.540 including the non-diagonalizable one. 00:32:44.540 --> 00:32:47.060 That -- that's the point, that in some way, 00:32:47.060 --> 00:32:50.430 Jordan completed the diagonalization by coming 00:32:50.430 --> 00:32:54.890 as near as he could, which is his Jordan form. 00:32:54.890 --> 00:32:57.340 And therefore, if you want to cover all matrixes, 00:32:57.340 --> 00:32:59.720 you've got to get him in the picture. 00:32:59.720 --> 00:33:03.030 It used to be -- when I took eighteen oh six, 00:33:03.030 --> 00:33:07.530 that was the climax of the course, this Jordan form stuff. 00:33:07.530 --> 00:33:11.490 I think it's not the climax of linear algebra anymore, 00:33:11.490 --> 00:33:15.930 because -- 00:33:15.930 --> 00:33:20.590 it's not easy to find this Jordan form 00:33:20.590 --> 00:33:25.340 for a general matrix, because it depends on these eigenvalues 00:33:25.340 --> 00:33:27.900 being exactly the same. 00:33:27.900 --> 00:33:30.620 You'd have to know exactly the eigenvalues and it -- 00:33:30.620 --> 00:33:35.270 and you'd have to know exactly the rank and the slightest 00:33:35.270 --> 00:33:39.180 change in numbers will change those eigenvalues, 00:33:39.180 --> 00:33:43.320 change the rank and therefore the whole thing is numerically 00:33:43.320 --> 00:33:46.970 not an -- a good thing. 00:33:46.970 --> 00:33:51.180 But for algebra, it's the right thing 00:33:51.180 --> 00:33:52.660 to understand this family. 00:33:52.660 --> 00:33:56.600 So just tell me another matrix -- a few more matrixes -- 00:33:56.600 --> 00:34:03.380 so more members of the family. 00:34:06.510 --> 00:34:11.380 Let me put down again what the best one is. 00:34:11.380 --> 00:34:12.400 Okay. 00:34:12.400 --> 00:34:13.409 All right. 00:34:13.409 --> 00:34:14.690 Some more matrixes. 00:34:14.690 --> 00:34:17.659 Let's see, what I looking for? 00:34:17.659 --> 00:34:22.750 I'm looking for matrixes whose trace is what? 00:34:22.750 --> 00:34:25.190 So if I'm looking for more matrixes in the family, 00:34:25.190 --> 00:34:28.290 they'll all have the same eigenvalues, four and four. 00:34:28.290 --> 00:34:30.449 So their trace will be eight. 00:34:30.449 --> 00:34:34.120 So why don't I just take, like, five and three -- 00:34:34.120 --> 00:34:40.540 I've got the trace right, now the determinant should be what? 00:34:40.540 --> 00:34:41.469 Sixteen. 00:34:41.469 --> 00:34:45.370 So I just fix this up -- shall I put maybe a one and a minus one 00:34:45.370 --> 00:34:46.771 there? 00:34:46.771 --> 00:34:47.270 Okay. 00:34:47.270 --> 00:34:52.139 There's a matrix with eigenvalues four and four, 00:34:52.139 --> 00:34:56.719 because the trace is eight and the determinant is sixteen. 00:34:56.719 --> 00:35:00.670 And I don't think it's diagonalizable. 00:35:00.670 --> 00:35:03.800 Do you know why it's not diagonalizable? 00:35:03.800 --> 00:35:06.470 Because if it was diagonalizable, 00:35:06.470 --> 00:35:09.505 the diagonal form would have to be this. 00:35:12.460 --> 00:35:14.980 But I can't get to that form, because whatever 00:35:14.980 --> 00:35:17.880 I do with any M inverse and M I stay with that form. 00:35:17.880 --> 00:35:20.320 I could never get -- connect those. 00:35:20.320 --> 00:35:22.470 So I can put down more members -- here -- 00:35:22.470 --> 00:35:23.810 here's another easy one. 00:35:23.810 --> 00:35:26.960 I could put the four and the four and a seventeen 00:35:26.960 --> 00:35:27.580 down there. 00:35:30.270 --> 00:35:32.080 All these matrixes are similar. 00:35:32.080 --> 00:35:35.350 If I'm -- I could find an M that would show that that one is 00:35:35.350 --> 00:35:37.350 similar to that one. 00:35:37.350 --> 00:35:40.420 And in -- you can see the general picture is I can take 00:35:40.420 --> 00:35:45.710 any a and any 8-a here and any -- oh, I don't know, 00:35:45.710 --> 00:35:48.200 whatever you put it'd be -- anyway, you can see. 00:35:48.200 --> 00:35:54.890 I can fill this in, fill this in to make the trace equal eight, 00:35:54.890 --> 00:36:01.160 the determinant equal 16, I get all that family of matrixes 00:36:01.160 --> 00:36:02.635 and they're all similar. 00:36:05.170 --> 00:36:08.250 So we see what eigenvalues do. 00:36:08.250 --> 00:36:13.020 They're all similar and they all have only one eigenvector. 00:36:13.020 --> 00:36:16.610 So I -- if I'm -- if you were going to -- 00:36:16.610 --> 00:36:20.780 allow me to add to this picture, they have the same lambdas 00:36:20.780 --> 00:36:25.799 and they also have the same number of independent 00:36:25.799 --> 00:36:26.340 eigenvectors. 00:36:29.880 --> 00:36:33.910 Because if I get an eigenvector for x I get one for -- for A, 00:36:33.910 --> 00:36:36.260 I get one for B also. 00:36:36.260 --> 00:36:43.170 So -- and same number of eigenvectors. 00:36:43.170 --> 00:36:45.740 But even more than that -- 00:36:45.740 --> 00:36:47.048 even more than that -- 00:36:47.048 --> 00:36:49.173 I mean, it's not enough just to count eigenvectors. 00:36:51.970 --> 00:36:54.450 Yes, let me give you an example why it's not even 00:36:54.450 --> 00:36:58.460 enough to count eigenvectors. 00:36:58.460 --> 00:37:00.170 So another example. 00:37:00.170 --> 00:37:04.210 So here are some matrixes -- 00:37:04.210 --> 00:37:07.600 oh, let me make them four by four -- 00:37:07.600 --> 00:37:09.260 okay, here -- here's a matrix. 00:37:09.260 --> 00:37:11.570 I mean, like if you want nightmares, 00:37:11.570 --> 00:37:14.480 think about matrixes like these. 00:37:14.480 --> 00:37:21.670 Uh, so a one off the diagonal -- say a one there, how many -- 00:37:21.670 --> 00:37:24.600 what are the eigenvalues of that matrix? 00:37:24.600 --> 00:37:29.040 Oh, I mean -- 00:37:29.040 --> 00:37:30.480 okay. 00:37:30.480 --> 00:37:32.550 What are the eigenvalues of that matrix? 00:37:35.740 --> 00:37:37.910 Please. 00:37:37.910 --> 00:37:40.160 Four 0s, right? 00:37:40.160 --> 00:37:44.930 So we're really getting bad matrixes now. 00:37:44.930 --> 00:37:47.080 So I mean, this is, like -- 00:37:47.080 --> 00:37:53.190 Jordan was a good guy, but he had to think about matrixes 00:37:53.190 --> 00:37:58.590 that all -- that had, like -- an eigenvalue repeated four times. 00:37:58.590 --> 00:38:01.140 How many eigenvectors does that matrix have? 00:38:04.750 --> 00:38:07.980 Well, I'm -- eigenvectors will be -- 00:38:07.980 --> 00:38:11.710 since the eigenvalue is zero, eigenvectors will be 00:38:11.710 --> 00:38:13.350 in the null space, right? 00:38:13.350 --> 00:38:17.870 I'm -- eigenvectors have got to be A x equal zero x. 00:38:17.870 --> 00:38:21.340 So what's the dimension of the null space? 00:38:21.340 --> 00:38:22.110 Two. 00:38:22.110 --> 00:38:23.920 Somebody said two. 00:38:23.920 --> 00:38:24.680 And that's right. 00:38:24.680 --> 00:38:26.690 How -- why? 00:38:26.690 --> 00:38:29.390 Because you ask what's the rank of that matrix, 00:38:29.390 --> 00:38:32.410 the rank is obviously two. 00:38:32.410 --> 00:38:35.050 The number of independent rows is two, 00:38:35.050 --> 00:38:36.880 the number of independent columns is two, 00:38:36.880 --> 00:38:41.430 the rank is two so the null -- the dimension of the null space 00:38:41.430 --> 00:38:45.950 is four minus two, so it's got two eigenvectors. 00:38:45.950 --> 00:38:47.500 Two eigenvectors. 00:38:47.500 --> 00:38:49.441 Two independent eigenvectors. 00:38:49.441 --> 00:38:49.940 All right. 00:38:49.940 --> 00:38:55.260 The dimension of the null space is two. 00:38:59.910 --> 00:39:03.905 Now, suppose I change this zero to a seven. 00:39:09.050 --> 00:39:11.840 The eigenvalues are all still zero, how -- what about -- 00:39:11.840 --> 00:39:12.756 how many eigenvectors? 00:39:15.449 --> 00:39:17.990 What's the dimension of the -- what's the rank of this matrix 00:39:17.990 --> 00:39:19.160 now? 00:39:19.160 --> 00:39:20.890 Still two, right? 00:39:20.890 --> 00:39:22.820 So it's okay. 00:39:22.820 --> 00:39:26.370 And actually, this would be similar to the one that 00:39:26.370 --> 00:39:27.800 had a zero in there. 00:39:27.800 --> 00:39:31.740 But it's not as beautiful, Jordan picked this one. 00:39:31.740 --> 00:39:34.850 He picked -- he put ones -- 00:39:34.850 --> 00:39:39.000 we have a one on the -- above the diagonal for every missing 00:39:39.000 --> 00:39:42.330 eigenvector, and here we're missing two because we've got 00:39:42.330 --> 00:39:45.550 two, so we've got two eigenvectors and two are 00:39:45.550 --> 00:39:53.860 missing, because it's a four by four matrix. 00:39:53.860 --> 00:39:58.910 Okay, now -- but I was going to give you this second example. 00:40:01.690 --> 00:40:05.145 0 1 0 0, let me just move the one. 00:40:09.030 --> 00:40:11.500 Oop, not there. 00:40:11.500 --> 00:40:15.710 Off the diagonal and zero zero zero zero zero. 00:40:15.710 --> 00:40:16.210 Okay. 00:40:19.420 --> 00:40:22.570 So now tell me about this matrix. 00:40:22.570 --> 00:40:27.460 Its eigenvalues are four zeroes again. 00:40:27.460 --> 00:40:31.910 Its rank is two again. 00:40:31.910 --> 00:40:36.670 So it has two eigenvectors and two missing. 00:40:36.670 --> 00:40:40.890 But the darn thing is not similar to that one. 00:40:40.890 --> 00:40:44.430 A -- a count of eigenvectors looks like these could be 00:40:44.430 --> 00:40:47.130 similar, but they're not. 00:40:47.130 --> 00:40:52.720 Jordan -- see, this is like -- a little three by three block 00:40:52.720 --> 00:40:55.880 and a little one by one block. 00:40:55.880 --> 00:40:58.770 And this one is like a two by two block and a two 00:40:58.770 --> 00:41:03.080 by two block, and those blocks are called Jordan blocks. 00:41:03.080 --> 00:41:06.770 So let me say what is a Jordan block? 00:41:11.150 --> 00:41:17.400 J block number I has -- 00:41:17.400 --> 00:41:22.660 so a Jordan block has a repeated eigenvalue, lambda I, lambda I 00:41:22.660 --> 00:41:24.530 on the diagonal. 00:41:24.530 --> 00:41:26.905 Zeroes below and ones above. 00:41:30.060 --> 00:41:34.160 So there's a block with this guy repeated, 00:41:34.160 --> 00:41:37.070 but it only has one eigenvector. 00:41:37.070 --> 00:41:40.900 So a Jordan block has one eigenvector only. 00:41:43.970 --> 00:41:47.720 This one has one eigenvector, this block has one eigenvector 00:41:47.720 --> 00:41:49.580 and we get two. 00:41:49.580 --> 00:41:52.170 This block has one eigenvector and that block has 00:41:52.170 --> 00:41:54.980 one eigenvector and we get two. 00:41:54.980 --> 00:42:02.020 So -- but the blocks are different sizes. 00:42:02.020 --> 00:42:05.540 And that -- it turns out Jordan worked out -- 00:42:05.540 --> 00:42:14.190 then this is not similar, not similar to this one. 00:42:18.020 --> 00:42:22.140 So the -- so I'm, like, giving you the whole story -- 00:42:22.140 --> 00:42:25.780 well, not the whole story, but the main themes of the story -- 00:42:25.780 --> 00:42:29.700 is here's Jordan's theorem. 00:42:29.700 --> 00:42:49.060 Every square matrix A is similar to A Jordan matrix J. 00:42:49.060 --> 00:42:51.890 And what's a Jordan matrix J? 00:42:51.890 --> 00:42:56.670 It's a matrix with these blocks, block -- 00:42:56.670 --> 00:43:03.850 Jordan block number one, Jordan block number two and so on. 00:43:03.850 --> 00:43:07.950 And let's say Jordan block number d. 00:43:11.360 --> 00:43:14.760 And those Jordan blocks look like that, 00:43:14.760 --> 00:43:17.760 so the eigenvalues are sitting on the diagonal, 00:43:17.760 --> 00:43:22.750 but we've got some of these ones above the diagonal. 00:43:22.750 --> 00:43:25.010 We've got the number of -- 00:43:25.010 --> 00:43:27.530 so the number of blocks -- 00:43:27.530 --> 00:43:37.510 the number of blocks is the number of eigenvectors, 00:43:37.510 --> 00:43:42.510 because we get one eigenvector per block. 00:43:42.510 --> 00:43:47.250 So what I'm -- so if I summarize Jordan's idea -- 00:43:47.250 --> 00:43:49.570 start with any A. 00:43:49.570 --> 00:43:54.080 If its eigenvalues are distinct, then what's it similar 00:43:54.080 --> 00:43:54.700 to? 00:43:54.700 --> 00:43:56.430 This is the good case. 00:43:56.430 --> 00:44:00.710 if I start with a matrix A and it has different eigenvalues -- 00:44:00.710 --> 00:44:03.450 it's n eigenvalues, none of them are repeated, 00:44:03.450 --> 00:44:09.350 then that's a diagonal -- diagonalizable matrix -- 00:44:09.350 --> 00:44:13.990 the Jordan blocks is -- has -- the Jordan matrix is diagonal. 00:44:13.990 --> 00:44:15.900 It's lambda. 00:44:15.900 --> 00:44:17.840 So the good case -- 00:44:17.840 --> 00:44:22.590 the good case, J is lambda. 00:44:27.020 --> 00:44:29.400 All -- there are -- 00:44:29.400 --> 00:44:29.900 d=n. 00:44:29.900 --> 00:44:33.830 There are n eigenvectors, n blocks, diagonal, everything 00:44:33.830 --> 00:44:35.640 great. 00:44:35.640 --> 00:44:40.990 But Jordan covered all cases by including 00:44:40.990 --> 00:44:44.460 these cases of repeated eigenvalues and missing 00:44:44.460 --> 00:44:47.100 eigenvectors. 00:44:47.100 --> 00:44:47.810 Okay. 00:44:47.810 --> 00:44:49.640 That's a description of Jordan. 00:44:49.640 --> 00:44:51.030 That -- that's -- 00:44:51.030 --> 00:44:54.300 I haven't told you how to compute this thing, 00:44:54.300 --> 00:44:56.500 and it isn't easy. 00:44:56.500 --> 00:45:00.920 Whereas the good case is the -- the good case is what 18.06 is 00:45:00.920 --> 00:45:01.660 about. 00:45:01.660 --> 00:45:07.040 The -- this case is what 18.06 was about 20 years ago. 00:45:07.040 --> 00:45:12.220 So you can see you probably won't have on the final exam 00:45:12.220 --> 00:45:18.540 the computation of a Jordan matrix for some horrible thing 00:45:18.540 --> 00:45:21.720 with four repeated eigenvalues. 00:45:21.720 --> 00:45:28.730 I'm not that crazy about the Jordan form. 00:45:28.730 --> 00:45:34.950 But I'm very positive about positive definite matrixes 00:45:34.950 --> 00:45:38.880 and about the idea that's coming Monday, 00:45:38.880 --> 00:45:40.964 the singular value decomposition. 00:45:40.964 --> 00:45:43.130 So I'll see you on Monday, and have a great weekend. 00:45:43.130 --> 00:45:44.680 Bye.